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Thursday, March 24, 2016

For nN,n2, prove that nk=1(12k112k)>2n3n+1.

For nN,n2, prove that
nk=1(12k112k)>2n3n+1.

Note that

112+1314+1516+12n112n

=112+1314+1516++12n112n

=(1+13+15++12n1)(12+14+16++12n)

=(1+12+13++12n)2(12+14+16++12n)

=(1+12+13++12n)(1+12+13++1n)

=(1+12+13++1n+1n+1+1n+2++12n)(1+12+13++1n)

=1n+1+1n+2++12n

Therefore

nk=1(12n112n)=1n+1+1n+2++12n(n times1+1++1)2n+1+n+2++2n(by the extended Cauchy-Schwarz inequality)=n2n2(n+1+2n)=2n3n+1(Q.E.D.)

4 comments:

  1. I love your posts keep them up! :)

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    1. Aww...thank you for your nice compliment!

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  2. Great job here Isabelle! Proud of you my friend. :-)

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  3. Aww!! This is a HUGE compliment from my dear friend, who speaks fluently in mathematics so well!:D

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